Weak gravity on a ΛcDM background

We consider Horndeski modified gravity models obeying stability, velocity of gravitational waves cT equals c and quasistatic approximation on subhorizon scales. We assume further a Lambda cold dark matter background expansion and a monotonic evolution on the cosmic background of the α functions as αi=αi0as where i=M, B, a is the scale factor and αi0 (αM0,αB0), s are arbitrary parameters. We show that the growth and lensing reduced (dimensionless) gravitational couplings μGgrowth/G, ςGlensing/G exhibit the following generic properties today: ς0<1 for all viable parameters, μ0<1 (weak gravity today) is favored for small s while μ0>1 is favored for large s. We establish also the relation μ≥ς at all times. Taking into account the fσ8 and EG data we constrain the parameter s to satisfy s≲2. Hence these data select essentially the weak gravity regime today (μ0<1) when s<2, while μ0>1 subsists only marginally for s≈2. At least the interval 0.5≲s≲2 would be ruled out in the absence of screening. We consider further the growth index γ(z) and identify the (αM0,αB0,s) parameter region that corresponds to specific signs of the differences γ0-γ0ΛCDM, and γ1-γ1ΛCDM, where γ0γ|z=0 and γ1dγdz|z=0. In this way important information is gained on the past evolution of μ. We obtain in particular the signature γ0>γ0ΛCDM for s<2 in the selected weak gravity region.